Observation of Rydberg-atom macrodimers: micrometer-sized diatomic molecules

Heiner Saßmannshausen and Johannes DeiglmayrLaboratory of Physical Chemistry, ETH Zurich, Switzerland∗

Long-range metastable molecules consisting of two cesium atoms in high Rydberg states havebeen observed in an ultracold gas. A sequential three-photon two-color photoassociation scheme wasemployed to form these molecules in states, which correlate to np(n + 1)s dissociation asymptotes.Spectral signatures of bound molecular states are clearly resolved at the positions of avoided crossingsbetween long-range van der Waals potential curves. The experimental results are in agreement withsimulations based on a detailed model of the long-range multipole-multipole interactions of Rydberg-atom pair states. We show that a full model is required to accurately predict the occurrence of boundRydberg macrodimers. The macrodimers are distinguished from repulsive molecular states by theirbehavior with respect to spontaneous ionization and possible decay channels are discussed.

A new regime of ultracold chemistry has been estab-lished by the possibility to routinely produce ultracoldatomic samples with translational temperatures below1 mK using laser cooling [1, 2] and trapping [3, 4]. Thecharacteristic feature of this regime is that reactions arenot driven by thermodynamical quantities, i.e. temper-ature or pressure, but by the precise manipulation of theinternal quantum states of the constituents and the inter-actions between them. Landmark results in the emerg-ing field of ultracold chemistry include the formation ofultracold molecules in the absolute ground state usingphotoassociation [5–7] or magneto-association [8, 9], thecontrol of chemical reactions by manipulating the quan-tum statistics of the reactants [10, 11], or the photodis-sociation of molecules with full control over reactant andproduct channels [12]. The enabling factor in all thesestudies is the control of long-range interaction potentials,which dominate the dynamics at low collision energies.

An extreme case of long-range interactions are van derWaals forces between two atoms in Rydberg states, scal-ing in the case of an off-resonant dipole-dipole interac-tion with the internuclear separation R as R−6 and withthe principal quantum number n as n11 [13]. Metastablemolecular states with internuclear separations exceeding1 µm, so called macrodimers, are predicted to resultfrom these interactions [14–16]. The molecular poten-tial minima supporting these bound states arise fromavoided crossings between long-range potential-energycurves correlating to different dissociation asymptotes ofthe doubly-excited dimers (see Fig. 1). Astonishingly,such macrodimers are predicted to have lifetimes lim-ited by radiative decay of the constituent atomic Ryd-berg states [17], even though they are energetically lo-cated in the Cs + Cs+ + e− ionization continuum andthe density of electronic states is extremely high, twoconditions one would expect to lead to fast autoioniza-tion [18]. Experimentally, these molecular states haveremained elusive. Molecular resonances were observed inthe Rydberg-excitation spectrum of ultracold gases andwere identified as arising from the correlated excitationof two interacting Rydberg atoms [19–22]. However, evi-dence for bound Rydberg-atom macrodimers could only

FIG. 1. (Color online) Left panel: Potential-energy curvesof two Cs Rydberg atoms in the vicinity of the 43p3/244s1/2dissociation asymptote. Black, red, and blue curves corre-spond to the molecular states with Ω=0,1, and 2, respec-tively. The intensity of the color codifies the value of the43p3/244s1/2 character (gray corresponding to zero, full colorto more than 0.5 % overlap with the asymptotic pair state).Bound macrodimer states are labeled by letters a to d. Rightpanel: the states a (lower curve) and b (upper curve) markedin the left panel are shown on a magnified scale. Energies(dashed lines) and probability densities of selected vibrationaleigenstates are indicated.

be obtained indirectly from dynamics following delayedpulsed-field ionization (PFI) for molecular states in aparticular electric field [20], and from the dynamics dur-ing the Penning ionization of excited atom pairs [22]. Inthis Letter, we present the observation of Rydberg-atommacrodimers by the unambiguous assignment of photoas-sociation resonances to long-range potential minima, cal-culated on the basis of a detailed model of the long-rangeinteractions [22]. An investigation of the lifetime of themacrodimers and a discussion of possible decay channelssupport the assignment.

The investigated macrodimer states were first identi-fied in an extensive theoretical search for minima in thelong-range potential functions of Cs Rydberg-atom pair

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FIG. 2. (Color online) a) Energy-level diagram for the two-color three-photon photoassociation of ground-state atomsto a doubly-excited molecular Rydberg state Ψ(R), b) se-quence of laser excitation and detection by pulsed-field ion-ization (PFI), c) sketch of the two-step excitation process,where black and red dots represent ground-state and Rydbergatoms, respectively, and the red circle illustrates the extentof the vibrational wavefunction of the macrodimer.

states, correlating to atomic ns, np, and nd Rydbergstates. The employed model is based on the multipoleexpansion of the static Coulomb interaction (truncatedafter terms corresponding to octupole-octupole interac-tions) and a basis constructed of atomic orbitals [22],where the only parameters are the atomic quantum de-fects taken from [23]. Contributions from electronic-parity-violating terms such as the dipole-quadrupole in-teraction, enabled by a coupling of quasi-degenerate ro-tational levels, are included in an effective manner [21].The only remaining molecular symmetry is associatedwith the projection of the total electronic angular mo-mentum on the internuclear axis with quantum numberΩ. The model is valid for interatomic distances larger

than the LeRoy radius RLR = 2(〈r2

A〉1/2

+ 〈r2B〉

1/2)

[24],

which is ¡ 0.6 µm for the states considered in this Letter.The identification of possible candidates for an experi-mental investigation was based on the following criteria:i) The equilibrium internuclear distance of the boundstate exceeds 1µm, the most likely next-neighbor dis-tance at a ground-state-atom density of 1011 cm−3, ii)the vibrational frequency exceeds 1 MHz which allows tospectrally resolve vibrational levels, and iii) the spectralsignatures of the bound states are clearly isolated fromother resonances. The most promising candidates for theobservation of macrodimer states in Cs under these re-strictions are states correlated to np3/2(n + 1)s1/2 pair-state asymptotes, with n = 43, 44 (see Fig. 1). An impor-tant finding of this theoretical study is that general pre-dictions for the existence of macrodimer states by moresimplified models (e.g. Ref. [25] applied to Cs atoms inzero field and Ref. [26] for Rb) could not be confirmed inour extended model.

The experimental scheme for the formation of

macrodimers by photoassociation is based on a sequen-tial excitation of the doubly excited molecular state (seeFig. 2) [27]. In a first step, a small number of atomsis resonantly excited into (n + 1)s1/2 Rydberg states.These atoms, which are referred to as “seed” Rydbergatoms in the following, modify the Rydberg-excitationspectrum of ground-state atoms in their vicinity. Theresulting interaction-induced shifts ∆ (see Fig. 2a)) sup-press the excitation of further atoms within a certainvolume around the seed atom and are the basis of theso-called dipole blockade mechanism [28]. The accessi-ble interaction potentials (Fig. 1), averaged over the dis-tribution of internuclear separations, can be probed byapplying a second, detuned laser pulse and monitoringthe additional excitation of ground-state atoms [29]. Toform macrodimers, the frequency of the second excitationlaser is tuned into resonance with the photoassociationtransition to the molecular state, which is detuned by ∆from the transition to the isolated np3/2 Rydberg state.A seed and a ground-state atom can then be photoas-sociated into a macrodimer when their scattering wave-function overlaps with the excited molecular wavefunc-tion (Fig. 2c)) [30]. It is noteworthy that this processis a true two-body transition in which the state of bothatoms is changed, similar as in laser-induced collisionalenergy transfers (LICET) [31].

The experiments are performed with dense (1 ·1012 cm−3), ultracold (T ≈ 40 µK) samples of Cs atoms,released from a crossed optical dipole trap. Only atomsin the densest part of the sample are selected for Ryd-berg excitation by optical pumping into the F = 4 hy-perfine component of the ground state. Details of theexperimental setup can be found in Refs. [22, 32]. Thens1/2 ← 6s1/2 two-photon transition of the excitationscheme outlined in Fig. 2 is driven by the pulse-amplifiedoutput of a diode laser (Toptica DL-100 pro), yieldingpulses with a duration of 7 ns and a near-transform-limited bandwidth of 60 MHz at wavelengths around639 nm. The photoassociation transition is driven by thecontinuous-wave frequency-doubled output of a ring dyelaser (Coherent 899 and MDB-200) with a wavelength of319 nm and a bandwidth of 1.5 MHz, from which UVlaser pulses of typically 1-10 µs duration and peak pow-ers exceeding 200 mW are obtained using an acousto-optical modulator. The frequencies of both lasers are sta-bilized to a commercial wavemeter (HighFinesse WS-7)referenced to a frequency comb (Menlo Systems FC1500-250-WG) [33]. Both laser beams are focused to spotsizes of ∼ 150µm in diameter and overlapped spatiallyat right angles in the Cs atom cloud (there is no tem-poral overlap between the two laser pulses). Transi-tions are detected by extracting Cs+ ions formed spon-taneously or by PFI of Rydberg atoms or molecules witha rising electric field (rise time ∼ 1 µs), and monitoringthe resulting ion yield using a microchannel-plate detec-tor. Spontaneously-formed ions and field-ionized Ryd-

3

FIG. 3. (Color online) Ion signals (black dots) as a func-tion of the detuning from the transition 43p3/2 ← 6s1/2: (a)Spontaneous-ion signal when no seed atoms are present, (b)PFI signal when 44s1/2 seed Rydberg atoms are present dur-ing excitation, and (c) ratio of spontaneous-ion to PFI signalswhen 44s1/2 seed Rydberg atoms are present (the backgroundsignal shown in panel a) was subtracted). The red line showsa simulated spectrum for a linewidth of 7 MHz full widthat half maximum, as discussed in the text. Labels a and bindicate the frequencies of transitions to the minima of themacrodimer potentials marked by a and b in Fig. 1.

berg atoms in different quantum states are discriminatedby their time of flight to the detector. Stray electric andmagnetic fields are compensated to below 25 mV/cm and20 MG, respectively, where their influence on the Ryd-berg energy spectrum is negligible for the relevant states.

Experimental spectra recorded below the 43p3/244s1/2

dissociation asymptote are presented in Fig. 3. Fig. 3a)shows the spontaneous-ion signal, extracted after a de-lay of 5 µs, as a function of the detuning of the UVlaser from the atomic 43p3/2 ← 6s1/2 transition with noseed excitations present (UV pulse duration 5 µs). A setof broadened resonances at detunings around −500 MHzis attributed to the excitation of long-range Rydbergmolecules bound by electron-atom scattering [34]. Theincreasing signal for higher frequencies is attributedto single-color two-photon excitation of Rydberg-atompairs. The spectra shown in Fig. 3b) and c) were recordedwhen about 50 seed excitations (corresponding to a den-sity of less than 5 ·107 cm−3) were present during the UVlaser pulse. The ion signal resulting from PFI of 44s1/2

seed Rydberg atoms is presented in Fig. 3b). The fluctu-ations of this signal (on the order of 5 ions after averagingover 30 laser-excitation and detection cycles) are causedby intensity instabilities of the pulse-amplified laser andmask additional excitations by the UV laser. As in pre-

vious works, we observe that interacting Rydberg-atompair states in the relevant range of detunings and dis-tances ionize within a few µs, leading to the appearanceof Cs+ ions [21, 22]. Surprisingly, the photoassociatedmacrodimers are also found to ionize within 5 µs. Theratio of spontaneous-ion to PFI signal shown in Fig. 3c) istherefore taken as a measure of the number of additionalexcitations per seed atom. In comparison to the back-ground measurement (Fig. 3a)), several additional reso-nances are present.

In order to interpret the experimental observations,we simulate the spectrum on the basis of the calculatedpotential-energy curves (see Fig. 1) [21, 22]. Assuminga resonant single-photon photoassociation, the expectedadditional signal at laser frequency ν = E/h is propor-tional to

s(E) =∑Ψ

∫ ∞0

ω2atomG(E − EΨ(R))pΨ(R)R2dR, (1)

where ωatom is the Rabi frequency for the atomic np3/2 ←6s1/2 transition, G(E) is the laser line profile, for whicha Gaussian function was chosen, and pΨ(R) is the sumof the squared np3/2(n+ 1)s1/2 coefficients of the molec-ular state Ψ with energy EΨ(R). This model is validwhen the mean number of additional excitations per seedatom is much smaller than 1, and the relative motion ofthe atoms can be treated classically. The latter assump-tion holds because the extent of the vibrational wave-functions of the macrodimers (see Fig. 1, right panel)is large compared to the thermal de Broglie wavelengthof the atoms (λth ∼ 30 nm). The simulated spectrum,shown as a red curve in Fig. 3c), reproduces all features ofthe experimental spectrum and allows for an unambigu-ous assignment of the observed resonances to molecularstates. The resonances at detunings of −530(3) MHzand −507(3) MHz arise from the potential minima of themacrodimer states with Ω = 0 and 1, labeled by lettersa and b, respectively, in Figs. 1 and 3. The broad reso-nance centered at −430 MHz results from avoided cross-ings of molecular states with Ω = 2. The resonances be-low −600 MHz arise from the maxima of potential-energycurves that result from the same avoided crossings lead-ing to the existence of the macrodimer states. For de-tunings above −480 MHz, the model underestimates thefraction of additional excitations. We attribute this de-viation to increasing contributions from an excitation-avalanche mechanism as discussed in Refs. [35–39]. Theexperimental spectra are reproduced best by a line profileG(E) with a width of 7 MHz, which is much larger thanthe measured linewidth of 2 MHz for an atomic tran-sition. The additional broadening is very likely causedby the thermal distribution of relative velocities in theatomic sample [40] (contributing about 1 MHz) and bythe interactions with all other seed atoms. The lattercontribution is estimated to be 3 MHz from the mea-sured broadening of the 43p3/2 ← 6s1/2 transition in the

4

FIG. 4. (Color online) Ratio of spontaneous-ion to PFI sig-nals as function of the detuning of the UV laser (2µs pulselength) from the 44p3/2 ← 6s1/2 transition when 45s1/2 seedatoms are present. Ions were extracted after a delay of 0 (graypoints) and 12 µs (black points). The spectrum simulated fora linewidth of 5 MHz (vertically scaled and offset) is shown assolid red and gray lines for comparison. The inset shows themeasured ratios (circles, squares, triangles, and diamonds) asa function of the extraction delay at the fixed laser detun-ings marked by the corresponding symbol on the frequencyaxis. A time-dependent background signal (measured at thefrequency indicated by an open circle) was subtracted fromall curves. Solid curves show the fit of a rate-equation modelto the data, the light-blue shaded region indicates the UV-excitation pulse.

presence of seed atoms with similar densities as in themeasurements presented in Fig. 3. These broadeningsmask the discrete resonance structures for photoassoci-ation into vibrational levels with a spacing of 1.3 MHz(see Fig. 1).

Photoassociation resonances of bound macrodimersare also observed on the low-frequency side of the44p3/245s1/2 dissociation asymptote, as shown in Fig. 4.The dependence of the spontaneous-ion ratio on theextraction delay reveals clear differences between reso-nances arising from the photoassociation of macrodimers,and from the excitation of Rydberg-atom pairs in thecontinuum. The spectra in Fig. 4 were recorded after ex-citation of about 50 atoms to the 45s1/2 state, followedby a 2-µs-long UV pulse. The spectrum presented inblack was obtained by extracting ions after a delay of12µs following UV excitation, whereas no delay was in-troduced in the measurement depicted in gray. The spec-tra exhibit an overall, UV-frequency-independent off-set that originates from spontaneous ionization of seedatoms. While the exact ionization mechanism is un-clear, the observed rate of 1.4 kHz is compatible with

calculations for black-body-enhanced field ionization ofns1/2 Cs Rydberg states [41]. More interestingly, theresonances originating from bound states at detuningsof −450, −425, and −355 MHz are clearly visible af-ter an extraction delay and are not present, or muchweaker, in the spectrum recorded without delay. Res-onances originating from potential maxima, e.g. at −583and −507 MHz, are observed with similar strengths un-der both measurement conditions. These observationsindicate that the bound macrodimers decay more slowlyinto Cs+ ions than Rydberg-atom pairs excited to contin-uum states. The inset of Fig. 4 depicts measurements ofthe background-corrected spontaneous-ion ratios Rcorr asa function of the extraction delay for different UV-laserdetunings together with results of a rate-equation modelfor the decay and formation processes of seed atoms andmacrodimers. The free parameters of this model, thephotoassociation and ionization rates at a given detun-ing, are determined from a fit of the model to the data.The model reveals that the apparent increase of Rcorr ob-served for all detunings results from the normalization tothe PFI signal, which decreases with time due to radia-tive decay (and to a much lesser extent due to ionization)of the initial seed atoms. The fitted ionization rates ex-ceed 3 MHz at the positions of the resonances locatedat detunings of −583 MHz and −507 MHz. Significantlylower ionization rates of about 0.6(1) and 1.0(1) MHzare obtained for the resonances at detunings of −421and −355 MHz, respectively, which correspond to boundmacrodimers. Although the corresponding lifetimes ofless than 2 µs are much shorter than expected [14, 17],they are still considerably longer than at the positions ofpotential maxima, which we consider a further indicationof the bound nature of these states.

Whereas the decay of Rydberg-atom pairs excited toattractive potentials can be understood in a Penning-ionization model [22, 42], the calculated potentials pre-sented in Fig. 1 indicate that such a process cannot occurfor the bound macrodimers. Other conceivable ionizationmechanisms are i) electronic transitions related to inter-Coulombic decay, in which one atom is transferred intothe (ionization) continuum while the other atom is de-excited into a more strongly bound state [43, 44], andii) vibrational autoionization, where the decay of thebound macrodimer is caused by non-adiabatic couplingsto molecular states with open ionization channels [18].By extending the models for mechanism i) and ii) de-scribed in Ref. [44] and Ref. [26], respectively, to higher-order interaction terms and larger basis sets, we verifiedthat neither mechanism can explain decay rates exceed-ing 1 kHz for the macrodimer states investigated here,even for vibrationally excited levels. The rates for ii) are,however, expected to increase significantly when interac-tions with a third atom, breaking the symmetry associ-ated with Ω and further increasing the density of accessi-ble rovibronic states, become important. The increased

5

linewidth of the molecular resonances discussed above in-dicates that the interaction with other seed atoms mightbe the reason for the observed fast decay rates.

In summary, we find that while the treatmentof Rydberg-Rydberg interactions on the basis of themultipole-expansion of the static Coulomb interactionyields very accurate adiabatic potential-energy functions,the dynamical evolution of the formed macrodimersclearly reveals the importance of non-adiabatic couplingsand few-body interactions for strongly correlated Ryd-berg systems. Future studies with single atoms in opticaldipole traps with tunable separations [45, 46], or atomsin an optical lattice [47], should enable the investigationof the importance of few-body interactions and probe theisolated dimer with the prospect of resolving the vibra-tional structure of macrodimers.

This work is supported financially by the ETH Re-search Grant ETH-22 15-1, the Swiss National ScienceFoundation under Project Nr. 200020-159848, the NCCRQSIT of the Swiss National Science Foundation, and theEU Initial Training Network COHERENCE under grantFP7-PEOPLE-2010-ITN-265031. We acknowledge theEuropean Union H2020 FET Proactive project RySQ(grant N. 640378). We thank Prof. Frederic Merkt (ETHZurich) for critical reading of the manuscript and for con-tinuous and exemplary support. The authors acknowl-edge fruitful discussions with Prof. Pierre Pillet (Lab-oratoire Aim Cotton, Orsay) and JD acknowledges in-structive discussions with PD Alexander Kuleff and Dr.Kirill Gokhberg (both Institute of Physical Chemistry,University of Heidelberg).

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